Finite Group Databasebeta

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#72313331

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

2921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8);(1,2)(3,6)(4,5)(7,8)(9,10,12,16)(11,19)(13,25,21,17)(14,18,24,30)(15,22)(20,32)(23,28)(26,39,36,31)(27,37)(29,38)(33,35)(34,40);(2,5)(6,8)(11,20,23,35)(13,26)(14,24)(15,27,29,40)(17,31)(18,30)(19,32,28,33)(21,36)(22,37,38,34)(25,39);(2,5)(3,7)(9,11,21,35,14,22,36,37,12,23,13,20,24,38,26,34)(10,15,25,40,18,28,39,33,16,29,17,27,30,19,31,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^10|b^2=f,c^2=fj,d^2=hj,e^2=g,h^2=i,i^2=j,c^b=cg,d^b=dgij,d^c=dh,e^a=eg,e^d=eg,f^d=fj,h^b=hj,h^c=hi,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);